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<h2 align=center>Ellipse Exhibit</h2>
<h4 align=center>(Plane Curves)</h4>

<h3>Brief Description:</h3>
<p> If a and b are two positive real numbers, with a greater than or 
  equal to b, then the standard ellipse in the Cartesian plane, having 
  semi-major axis length a and semi-minor axis length b, is given 
  parametrically by the equations x = a * cos(t) and y = b * sin (t), 
  with t between 0 and 2 pi, or implicitly by the equation 
  (x/a)^2 + (y/b)^2 = 1. </p>
<p>The eccentricity of the ellipse is 
  the non-negative real number  e = sqrt( 1 - (b/a)^2).
  Note that if a = b = r, then the ellipse is the circle of radius r 
  centered at the origin. Thus a circle is the special case of an 
  ellipse for which the eccentricity is 0. </p>
<p>In polar coordinates (r,theta), 
  if l = a * (1 - e^2) is the so-called semi-latus rectum, then the 
  equation of the ellipse is  r * (1 + e * cos(theta)) = l.
  (In classical mechanics, this is the form in which one finds the ellipse
  when one derives Kepler's famous "First Law"---that the planets travel in 
  ellipses with the Sun at one focus---from Newton's Laws of Motion and 
  Newtons Law of Gravitation---that the force the Sun exerts on a planet 
  is directed toward the Sun and its magnitude is proportional to the 
  reciprocal of the square of the distance from the Sun to the planet.  </p>
<h3>What You Will See When You Select the Exhibit:  </h3>
<p>You will see an ellipse with the its foci marked by crosses, and 
  one of them labelled "F". Call the second (unmarked) vertex F'. There 
  is a blue circle with center F' and with radius the major-axis length L. 
  A point marked "S" moves around this circle; call the point where the 
  radius through S meets the ellipse "Q". (It is not marked in the Exhibit.)
  The distances QS and QF stay visibly equal, indicating that the ellipse 
  is the locus of points equidistant from F and the circle. Another segment 
  is drawn from Q to F, and as Q moves around the ellipse, it should be 
  reasonably evident the sum of their lengths |QF| + |QF'| is L, i.e., 
  the ellipse is also the locus of points the sum of whose distances 
  from F and F' is L. Moreover these two segments, QF and QF', make 
  equal angles with the (green) tangent line to the ellipse at Q, 
  illustrating that a light ray emitted at one focus and reflecting off 
  the ellipse will pass though the other vertex.  </p>
<h3>More Details:  </h3>
<p>A classic (and ancient) definition of an ellipse is as follows:
  Choose two points F and F' in a plane P; these will be the two 
  so-called foci of the ellipse (each is called a focus). Also choose 
  a positive real number L, greater than the distance between the two foci. 
  Then the ellipse E with foci F and F' and major axis length L is the 
  locus of points in the plane P for which the sum of its distances 
  from the two foci equals L. Intuitively speaking, place pins 
  at F and F', and tie a piece of string of length L joining them. 
  Now use a sharp pencil point to stretch the string taut, and trace out 
  all the points that the pencil can reach. The eccentricity of the 
  ellipse E is the ratio, e, of the distance |FF'| to the major 
  axis length L. Define a Cartesian coordinate system with 
  its origin at the midpoint of F and F' and with x-axis the line 
  joining them. Let a = L/2 be the semi-major axis length and let 
  b = a * sqrt(1 - e^2). Then the two foci are at (e * a, 0) and 
  (- e * a, 0) and the ellipse is given as above by (x/a)^2 + (y/b)^2 = 1. </p>
<p>A less familiar but equivalent geometric definition of an ellipse is as 
  follows. Draw a circle of radius L centered at one focus, say F'. 
  Then the ellipse E is the locus of points Q such that the distance 
  of Q to the other focus, F, equals the distance of Q from the circle C.  </p>
<p>Still another geometric definition of the ellipse is 3-dimensional in 
  nature. It says that an ellipse in the plane P can always be realized 
  as the intersection of P with a right circular cone K with vertex v 
  outside the plane---in other words, an ellipse is a conic section.
  Another way to think of this is that an ellipse is the shadow on P of a 
  circle (a section of the cone) thrown by a light at the vertex v.  </p>
<p>There is also an interesting "rolling construction" of the ellipse. 
  Let a circle C1 of radius r roll along the inside of a circle C2 of 
  radius R. Choose a radius of C1 that moves along with it as C1 rolls 
  inside C2 and let Q be the point of distance d from the center along 
  this radius. Then the path traced out by Q is called a hypertrochoid. 
  A  remarkable fact is that if r = R/2, then the resulting hypertrochoid 
  is an ellipse. (The semi-major axis length is d + r and the semi-minor 
  axis length is d - r.)  </p>
<p>A useful properties of an ellipse, called the reflection property, 
  is that a ray of light emitted from one vertex and reflecting off the 
  ellipse will pass through the other vertex. This just means that if you 
  draw line segments from each focus to the same point Q on the ellipse, 
  then the normal to the ellipse at Q bisects the angle FQF', or 
  equivalently that QF and QF' make equal angles with the tangent to 
  the ellipse at Q.  This accounts for the "whispering gallery" effect 
  of an elliptically shaped room; a word whispered softly at one focus 
  can be heard clearly at the other focus, but not elsewhere in the room.  </p>
<h3>Things to Try:  </h3>
<p>Try the various items in the Action menu, and see About This Gallery 
  for the Plane Curve Gallery for a discussion of these items.  Also, 
  try Morphing the ellipse, using the Animate menu. </p>
<h3>Links to Further information:  </h3>
<p><a href="http://en.wikipedia.org/wiki/Ellipse">http://en.wikipedia.org/wiki/Ellipse  </a></p>
<p><a href="http://xahlee.org/SpecialPlaneCurves_dir/Ellipse_dir/ellipse.html">http://xahlee.org/SpecialPlaneCurves_dir/Ellipse_dir/ellipse.html </a></p>
<p><a href="http://www-groups.dcs.st-and.ac.uk/~history/Curves/Ellipse.html http://www.du.edu/~jcalvert/math/ellipse.htm">http://www-groups.dcs.st-and.ac.uk/~history/Curves/Ellipse.html
  http://www.du.edu/~jcalvert/math/ellipse.htm  </a></p>
<p><a href="http://www.ies.co.jp/math/java/conics/focus_ellipse/focus_ellipse.html">http://www.ies.co.jp/math/java/conics/focus_ellipse/focus_ellipse.html  </a></p>
<p><a href="http://www.daviddarling.info/encyclopedia/E/ellipse.html">http://www.daviddarling.info/encyclopedia/E/ellipse.html </a></p>


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